Bayesian Inverse Problems and Deep Learning

Inverse Problems are a fascinating branch of mathematics with strong connections to other areas such as (functional) analysis, numerical mathematics, optimization, stochastics, and many more. At the same time, inverse problems are close to applications in physics, medicine, biology, engineering and, again, many more. In fact, one may describe inverse problems as the task to determine a sought-after quantity throu indirect measurements, or, in other words, to determine a cause from its effect. Such problems are often ill-posed, and straight forward solution approaches fail.

Until recently, the main branches of inverse problems research were the classical deterministic approach, and the Bayesian, stochastic, approach. Recently, Machine Learning and Deep Learning have become the hot topics in current inverse problems research. As a consequence, no unified theory is available yet, and there are many opportunities for young researchers to contribute.

This lecture serves as an introduction to the current state of Learning approaches for inverse problems. To this end, classical results for deterministic and Bayesian inverse problems are presented and reviewed from a novel point of view. Then, currently available Learning methods are presented, which are typically extensions of classical ideas.

The course is designed to be understandable for master students. Previous knowledge of an inverse problems lecture is helpful, but not necessary.

The course language (English or German) will be determined in the first lecture depending on the audience.